Vals and tuning space: Difference between revisions

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{{interwiki

| de = Val

| en = Vals and ~~Tuning Space~~

| en = Vals and tuning space

| es =

| ja = ~~ヴァルと音程空間~~

| ja = ヴァルと調律空間

}}

~~__FORCETOC__~~

== Definition ==

A [[val]] "maps" just intonation to a certain number of steps in a chain of generators; by putting vals together we can define the mapping of a [[regular temperament]] and thereby define the temperament. A val is written in the form {{val| ''a''1 ''a''2 ''a''3 … ''a''k }}, where the numbers ''a''1 ''a''2 ''a''3 … are the number of steps along the chain that the first ''k'' primes are mapped to. This can be generalized so that ''a''1 ''a''2 ''a''3 … represent the number of steps any JI [[basis]] is mapped to, whereas a JI basis for a [[just intonation subgroup]] is an independent collection of just intonation intervals, meaning that no one of them is a product of the rest.

A [[val]] "maps" just intonation to a certain number of steps in a chain of generators; by putting vals together we can define the mapping of a [[regular temperament]] and thereby define the temperament. A val is written in the form {{val| ''a''1 ''a''2 ''a''3 … ''a''''k'' }}, where the numbers ''a''1 ''a''2 ''a''3 … are the number of steps along the chain that the first ''k'' primes are mapped to. This can be generalized so that ''a''1 ''a''2 ''a''3 … represent the number of steps any JI [[basis]] is mapped to, whereas a JI basis for a [[just intonation subgroup]] is an independent collection of just intonation intervals, meaning that no one of them is a product of the rest.

A ''rank-r'' temperament has ''r'' generators, and thus is defined by ''r'' vals. In the usual coordinates for the [[Harmonic limit|''p''-limit]], the set of generators are the first ''k'' prime numbers and the set of vals for a ''p''-limit temperament gives you the coordinates for each prime harmonic in the ''p''-limit. For example, all 5-limit rank-1 temperaments, or [[equal temperament]]s, will be defined by a val {{val| ''a'' ''b'' ''c'' }}, where ''a'' is the number of generators it takes to reach the 2nd harmonic (2/1), ''b'' is the number of generators to reach the 3rd harmonic (3/1), and ''c'' is the number of generators it takes to reach the 5th harmonic (5/1). All 5-limit rank-2 temperaments are defined by two vals: {{monzo| {{val| ''a''1 ''b''1 ''c''1 }}, {{val| ''a''2 ''b''2 ''c''2 }} }}. Now, we locate the 2nd harmonic (2/1) with the 2-dimensional coordinates (''a''1, ''a''2), sometimes written as {{monzo| ''a''1 ''a''2 }}, meaning go up ''a''1 of the first generator, and up ''a''2 of the 2nd generator, to reach 2/1. Similarly, the 3rd harmonic and 5th harmonic will be reached by {{monzo| ''b''1 ''b''2 }} and {{monzo| ''c''1 ''c''2 }} respectively.

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 {{interwiki
 | de = Val
-| en = Vals and Tuning Space
+| en = Vals and tuning space
 | es =
-| ja = ヴァルと音程空間
+| ja = ヴァルと調律空間
 }}
+{{Expert|Val}}
-__FORCETOC__
 == Definition ==
-A [[val]] "maps" just intonation to a certain number of steps in a chain of generators; by putting vals together we can define the mapping of a [[regular temperament]] and thereby define the temperament. A val is written in the form {{val| ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> … ''a''<sub>k</sub> }}, where the numbers ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> … are the number of steps along the chain that the first ''k'' primes are mapped to. This can be generalized so that ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> … represent the number of steps any JI [[basis]] is mapped to, whereas a JI basis for a [[just intonation subgroup]] is an independent collection of just intonation intervals, meaning that no one of them is a product of the rest.
+A [[val]] "maps" just intonation to a certain number of steps in a chain of generators; by putting vals together we can define the mapping of a [[regular temperament]] and thereby define the temperament. A val is written in the form {{val| ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> … ''a''<sub>''k''</sub> }}, where the numbers ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> … are the number of steps along the chain that the first ''k'' primes are mapped to. This can be generalized so that ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> … represent the number of steps any JI [[basis]] is mapped to, whereas a JI basis for a [[just intonation subgroup]] is an independent collection of just intonation intervals, meaning that no one of them is a product of the rest.
 A ''rank-r'' temperament has ''r'' generators, and thus is defined by ''r'' vals. In the usual coordinates for the [[Harmonic limit|''p''-limit]], the set of generators are the first ''k'' prime numbers and the set of vals for a ''p''-limit temperament gives you the coordinates for each prime harmonic in the ''p''-limit. For example, all 5-limit rank-1 temperaments, or [[equal temperament]]s, will be defined by a val {{val| ''a'' ''b'' ''c'' }}, where ''a'' is the number of generators it takes to reach the 2nd harmonic (2/1), ''b'' is the number of generators to reach the 3rd harmonic (3/1), and ''c'' is the number of generators it takes to reach the 5th harmonic (5/1). All 5-limit rank-2 temperaments are defined by two vals: {{monzo| {{val| ''a''<sub>1</sub> ''b''<sub>1</sub> ''c''<sub>1</sub> }}, {{val| ''a''<sub>2</sub> ''b''<sub>2</sub> ''c''<sub>2</sub> }} }}. Now, we locate the 2nd harmonic (2/1) with the 2-dimensional coordinates (''a''<sub>1</sub>, ''a''<sub>2</sub>), sometimes written as {{monzo| ''a''<sub>1</sub> ''a''<sub>2</sub> }}, meaning go up ''a''<sub>1</sub> of the first generator, and up ''a''<sub>2</sub> of the 2nd generator, to reach 2/1. Similarly, the 3rd harmonic and 5th harmonic will be reached by {{monzo| ''b''<sub>1</sub> ''b''<sub>2</sub> }} and {{monzo| ''c''<sub>1</sub> ''c''<sub>2</sub> }} respectively.